Review
Chapter 5 (Trigonometric
Functions of Real Numbers)
The Unit Circle:
Is the circle of radius 1 centered at the origin in the xy-plane.
The equation is: x 2  y 2  1 .
*Any point ( x, y ) on the unit circle will satisfy the equation of the unit circle
*When given a partial point (x, # ) or (# , y ) you can plug point into
equation of the unit circle and find missing x or y value.
Examples of Problems with The Unit Circle:

3 1
,  on the unit circle.
(1)
Is the point  
2
2

 5 
, y  is on the unit circle in quadrant 4. Find its y(2)
The point P
6


coordinate.
Radian Measure: If a circle of radius 1 is drawn with the vertex of an angle at
its center, then the measure of this angle in radians is the length of the arc
that subtends the arc.
r
r
Terminal Points of the Unit Circle:
Suppose t is a real number. Lets mark off a distance t along the unit
circle, starting at the point (1,0) and moving in a counterclockwise
direction if t is positive or in a clockwise direction if t is negative. In this
way we arrive at a point P ( x, y ) on the unit circle. The point P ( x, y )
obtained in this way is called the terminal point determined by the real
number t .
Special Terminal Values:
t
Terminal point determined by t
0

6
(1,0)
 3 1


 2 ,2



4
 2 2


 2 , 2 



3
1 3
 ,

2 2 



2
(0,1)
Reference Number: let t be a real number. The reference
number t associated with t is the shortest distance along the unit
circle between the terminal point determined by t and the x axis.
a

When given t  
, the reference number is t  
b
b
Using the Unit Circle Chart to Find Terminal Points:
To find the terminal point P determined by any value of t  
we use the following steps:

a
,
b
1)
Find the reference number t . (Remember t  
2a)
Depending on the sign of t , move t a time
(counterclockwise or clockwise)
a
You could reduce the fraction of t  
then follow 2a.
b
2b)
Examples of Problems with Terminal Points of the Unit Circle:
(1)
Find the terminal point determined by each given number t :
7
13
5
t
t
t
(a)
(b)
(c)
6
4
6
101
50
55
t
t
t
(d)
(e)
(f)
3
4
6
29
55
19
t
t
t
(g)
(h)
(i)
6
6
4
b
)
Trigonometric Functions:
Let t be any real number and let P( x, y ) be the terminal point on the unit
circle determined by t . We define:
y
tan t 
( x  0)
sin t  y
cos t  x
x
1
1 hyp
x
sec t 
( x  0)
csc t  
( y  0)
cot t 
( y  0)
x
y opp
y
(Note pairs of inverses:
sin and csc , cos and sec , tan and cot )
Special Values of Trigonometric Functions:
t
sin t
cos t
tan t
csc t
sec t
cot t
0

6

4

3

2
0
1
2
0
3
3
------------
1
------------
2
2 3
3
3
2
2
3
2
1
3
2
2
2
1
2
1
2
2
1
3
2 3
3
2
3
3
1
0
------------
1
------------
0
(It is most important to know cos t and sin t , the rest can be derived
from those)
Domains of the Trigonometric Functions:
Function
Domain
sin, cos
All real numbers
tan, sec
cot, csc

 n for any integer n
2
All real numbers other than n for any integer n
All real numbers other than
Values of trigonometric Functions:
The following diagram shows in
which quadrants the trig functions are positive
y
sin
csc
All
x
tan
cot
cos
sec
Reciprocal Identities:
sin t 
opp
hyp
csc t 
1
hyp

sin t opp
cos t 
adj
hyp
sec t 
1
hyp

cos t adj
tan t 
sin t opp

cos t adj
cot t 
cos t adj
1


tan t sin t opp
Pythagorean Identities:
sin 2 t  cos 2 t  1
tan 2 t  1  sec 2 t
1  cot 2 t  csc 2 t
Examples of Problems with Trigonometric Functions:
(1)
In what quadrant(s) are the following possible:
Sec t  0
(Cot t )( Sin t )  0
(a)
(b)
Tan t  0 and Cos t  0
( Sin t )(Cos t )(Tan t )  0
(c)
(d)
cos t  0 and cot t  0
(e)
(2)
Evaluate each of the trigonometric functions:
19
 3 
 23 
sin
tan  
sec 
(a)
(b)
(c)


4
3 
 4 

7
51
 
csc
tan
csc  
(d)
(e)
(f)
6
6
 4
3 4
(3)
The terminal point  ,  determined by a real number t . Find all six trig
5 5
functions. (make sure all fractions are reduced to lowest terms)
tan t sin t
(4)
What is the sign of
, in quadrant three?
cot t
4
(5)
Find the values of all trigonometric functions if cos t   and the terminal
5
point of t is in quadrant three. (Make sure all fractions are in lowest terms,
all radicals are in simplest radical form and there are no radicals in the
denominator)
Graphs of Trigonometric Functions:
Typical graph of sine:
Typical graph of cosine:
Trig Functions:
The sine and cosine curves
and
y  a sin k ( x  b)  m
y  a cos k ( x  b)  m
(k  0) if a value is negative, the graph is reflected over the
x axis.
a)
The number a is called the amplitude and is the largest
b)
c)
d)
e)
value these functions can obtain.
2
The number
is called the period and is the distance
k
between any two repeating points on the function.
Horizontal Shift b , means how far the graph is shifted left or
right
Vertical Shift m , means how far the graph is shifted up or
down
An appropriate period on which to graph one complete

 2 
period is b, b  

 k 

Examples of Problems with Graphs of Trigonometric Functions:
(1)
Find the amplitude, period and phase shift of each, also graph one period
of the function:

3
2 


y  3 sin 2 x  
y  cos  2 x 
(a)
(b)

4
4
3 



4 


y  2 sin  x  
y  3 cos 2 x 
(c)
(d)

6
3 


(won’t have to graph trig function with a phase shift to the left)
y  a sin( k (t  b))  m or y  a cos( k (t  b))  m
Simple Harmonic Motion:
Where y represents the displacement of an object after time t , m
represents the value of a vertical shift and b represents the
value of a horizontal shift. Also:
Maximum displacement of the object
Amplitude  a
Period 
2
k
Frequency 
Time required to complete one cycle
k
2
Number of cycles per unit of time
Examples of Problems with Simple Harmonic Motion:
(1)
Vibrating String: The displacement of a mass suspended by a spring is
modeled by the function y  10 sin 4t where y is measured in inches and
t seconds.
(a)
Find the amplitude, period, and frequency of the motion of mass.
(b)
Sketch the graph of the displacement of the mass.
(2)
Vibrations of a Musical Note: A tuba player plays the note E and sustains
the sound for some time. For a pure E the vibration in pressure from
normal air pressure is given by V (t )  .2 sin 80t where V is measured in
pounds per square inch and t in seconds.
(a)
Find the amplitude, period and frequency of V .
(b)
Sketch a graph of V .
(c)
If the tuba player increases the loudness of the note, how much
does the equation for V change?
(d)
If the player is playing the note incorrectly and it is a little flat, how
does the equation for V change?
(3)
Modeling a Vibrating String:
A mass is suspended from a spring.
The spring is compressed a distance of 4 cm and then released. It is
1
observed that the mass returns to the compressed position after
3
seconds.
(a)
Find a function that models the displacement of the mass.
(b)
Sketch the graph of the displacement of the mass.
(4)
Modeling the Bright of a Variable Star: A variable star is one whose
brightness alternately increases and decreases. For the variable star
Delta Cephei, the time between periods of maximum brightness is 5.4
days. The average brightness (or magnitude) of the star is 4.0, and its
brightness varies by  0.35 magnitude.
(a)
Find a function that models the brightness of Delta Cephei as a
function of time.
(b)
Sketch a graph of the brightness of Delta Cephei as a function of
time.
Damped Harmonic Motion:
If the equation describing the displacement y of an object at time t is
( c  0)
y  ae  ct sin kt or y  ae ct cos kt
then the object is in damped harmonic motion. The constant c is the
2
damping constant, a is the initial amplitude, and
is the period.
k
Examples of Problems with Damped Harmonic Motion:
(1)
When a car hits a certain bump on the road, a shock absorber on the car
is compressed a distance of 6 in, then released. The shock absorber
vibrates in damped harmonic motion with a frequency of 2 cycles per
second. The damping constant for this particular shock is 2.8.
(a)
Find an equation that describes the displacement of the shock
absorber from its rest position as a function of time. (Take t  0 to
be the instant that the shock absorber is released)
(b)
How long does it take for the amplitude of the vibration to decrease
to 0.5 in.?
(2)
Two mass-spring systems are experiencing damped harmonic motion,
both at 0.5 cycles per second, and both with an initial maximum
displacement of 10 cm. The first has a damping constant of 0.5 and the
second has a damping constant of 0.1.
(a)
Find functions of the form g (t )  ae ct cos kt to model the motion in
each case.
(b)
Graph the two functions you found in part (a). How do they differ?
(3)
A stone is dropped in a calm lake, causing waves to form. The up-anddown motion of a point on the surface of the water is modeled by damped
harmonic motion. At some time the amplitude of the wave is measured,
1
and 20 seconds later it is found that the amplitude has dropped to
of
10
its value. Find the damping constant c .